When Ford-Lincoln-Mercury and the postal carrier from California (Clark) are about to be executed, and Ford introduces himself, we flash to a distance shot, in which we see Clark remove his hat. We then flash to a closeup, and Clark removes his hat again.
The main character assumes his role because he found an old abandoned mail truck containing the remains of a long dead mailman and pilfered the uniform from the skeleton. The problem here is the condition of the clothing he takes. When a person dies, the body goes through many stages of decomposition on its way to being merely a skeleton. As the tissues break down, many chemicals and enzymes are released, including the hydrochloric acid of the digestive system. In the final stages of decomposition, this is referred to as liquefacation or liquiescence. Given enough time and a suitable environment this combination of byproducts, with the addition of the bacteria that will inevitably emerge, would make any cloth or fabric (with the exception of treated leather products) not only disgustingly filthy, but also so weakened from exposure to what amounts to a corrosive liquid, that the fibers would tear apart from any stresses put on them. Even the act of taking the jacket off of the remains (and certainly that of putting the jacket on himself) would have pulled the fabric apart.